An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann–Morgenstern utility functions on a finite set of alternatives A. Each agent i  's utility function is known to lie in the nonempty, convex, relatively open set UiUi. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L   no matter what the true utility functions are. Then, there exist utility functions ui∈Uiui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.